is an associative algebra over some ground field k, then a left associative
called the source map, an algebra map
called the target map, so that the elements of the images of
which is required to be a counital coassociative comultiplication on
in the monoidal category of
is required to be a left character (equivalently, the map
must be a left action extending the multiplication
is not an algebra, hence asking for a bialgebra-like compatibility that
is a morphism of k-algebras does not make sense.
which contains the image of
and has a well-defined multiplication induced from its preimage under the projection from the usual tensor square algebra
Then one requires that the corestriction
is a homomorphism of unital algebras.
, one can make a canonical choice for
, namely the so called Takeuchi's product
,[2] which always inherits an associative multiplication via the projection from
Thus, it is sufficient to check if the image of
is contained in the Takeuchi's product rather than to look for other
As shown by Brzeziński and Militaru, the notion of a bialgebroid is equivalent to the notion of
-algebra introduced by Takeuchi earlier, in 1977.
[3] Associative bialgebroid is a generalization of a notion of k-bialgebra where a commutative ground ring k is replaced by a possibly noncommutative k-algebra
Hopf algebroids are associative bialgebroids with an additional antipode map which is an antiautomorphism of
satisfying additional axioms.
The term bialgebroid for this notion has been first proposed by J-H.
[4] The modifier associative is often dropped from the name, and retained mainly only when we want to distinguish it from the notion of a Lie bialgebroid, often also referred just as a bialgebroid.
Associative bialgebroids come in two chiral versions, left and right.
[5] There is a generalization, an internal bialgebroid which abstracts the structure of an associative bialgebroid to the setup where the category of vector spaces is replaced by an abstract symmetric monoidal category admitting coequalizers commuting with the tensor product.